reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

theorem
  R.:X \ R.:Y c= R.:(X \ Y)
proof
  let y be object;
  assume
A1: y in R.:X \ R.:Y;
  then consider x such that
A2: [x,y] in R and
A3: x in X by Def11;
  not y in R.:Y by A1,XBOOLE_0:def 5;
  then not x in Y or not [x,y] in R by Def11;
  then x in X \ Y by A2,A3,XBOOLE_0:def 5;
  hence thesis by A2,Def11;
end;
